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August, 2019 Admissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-line
Thieu Huy Nguyen, Xuan Yen Trinh
Taiwanese J. Math. 23(4): 897-923 (August, 2019). DOI: 10.11650/tjm/181209


For the following class of partial neutral functional differential equations \[ \begin{cases} \frac{\partial}{\partial t} Fu_t = B(t) u(t) + \Phi(t,u_t) &t \in (0,\infty), \\ u_0 = \phi \in \mathcal{C} := C([-r,0],X) \end{cases} \] we prove the existence of a new type of invariant stable and center-stable manifolds, called admissibly invariant manifolds of $\mathcal{E}$-class for the solutions. The existence of such manifolds is obtained under the conditions that the family of linear partial differential operators $(B(t))_{t \geq 0}$ generates the evolution family $\{U(t,s)\}_{t \geq s \geq 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|\Phi(t,\phi)-\Phi(t,\psi)\| \leq \varphi(t) \|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line. Our main method is based on Lyapunov-Perrons equations combined with the admissibility of function spaces and fixed point arguments.


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Thieu Huy Nguyen. Xuan Yen Trinh. "Admissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-line." Taiwanese J. Math. 23 (4) 897 - 923, August, 2019.


Received: 27 July 2018; Revised: 9 December 2018; Accepted: 17 December 2018; Published: August, 2019
First available in Project Euclid: 18 July 2019

zbMATH: 07088953
MathSciNet: MR3982067
Digital Object Identifier: 10.11650/tjm/181209

Primary: 34C45 , 34G20 , 35B40 , 37D10

Keywords: admissibility of function spaces , exponential dichotomy , Partial neutral functional differential equations , stable and center-stable manifolds

Rights: Copyright © 2019 The Mathematical Society of the Republic of China


Vol.23 • No. 4 • August, 2019
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